Basic formulation

Let us consider the propagation of cosh-Gaussian laser beams through collisionless magnetized plasma along the z direction, which is the direction of static magnetic field B ₀. The electric field of the laser beam propagating in either of the two modes, that is, extraordinary and ordinary can be written as,

(1)$$E_ \pm = \hat x\,E_{0 \pm} (r,z,t)\exp [ - i({\rm \omega} t - k_ \pm z)],$$

where $k_ \pm = {\rm \omega} /c\sqrt {{\rm \varepsilon} _{0 \pm}} $ is the propagation constant of the wave. Here ε_0± is the linear part of plasma dielectric constant and c is the speed of light in vacuum. The effective dielectric constant of magnetized plasma can be written as,

(2)$${\rm \varepsilon} _ \pm = {\rm \varepsilon} _{0 \pm} + {\rm \Phi} _ \pm (EE^{^\ast}),$$

where ${\rm \varepsilon} _{0 \pm} = 1 - {\rm \omega} _{\rm p}^2 /{\rm \omega} ({\rm \omega} \mp {\rm \omega} _{\rm c})$ is the linear part of dielectric constant with ω_p = (4πN ₀e ²/m)^1/2 as a plasma oscillation frequency and ω_c = eB ₀/mc as a cyclotron frequency. Here, e and m are the electronic charge and rest mass, respectively.

The second term in [Eq. (2)], the intensity-dependent part of dielectric constant for a collisionless magnetized plasma, is given by

(3)$${\rm \Phi} _ \pm (E_ \pm E_ \pm ^{^\ast} ) = \displaystyle{{{\rm \omega} _{\rm p}^2} \over {2{\rm \omega} ({\rm \omega} \mp {\rm \omega} _{\rm c})}}[1 - \exp ( - {\rm \alpha} E_ \pm E_ \pm ^{^\ast} )].$$

In the light of Maxwell's equations, the general form of wave equation governing the propagation of laser beam is given as,

(4)$$\displaystyle{{\partial ^2E_ \pm} \over {\partial z^2}} + {\rm \delta} _ \pm \left( {\displaystyle{{\partial ^2E_ \pm} \over {\partial r^2}} + \displaystyle{1 \over r}\displaystyle{{\partial E_ \pm} \over {\partial r}}} \right) + \displaystyle{{{\rm \omega} ^2} \over {c^2}}({\rm \varepsilon} _{_ \pm} E_ \pm ) = 0,$$

where ${\rm \delta} _ \pm = 1/2(1 \mp {\rm \varepsilon} _{0 \pm} /{\rm \varepsilon} _{0zz})$ and ${\rm \varepsilon} _{0zz} = 1 - {\rm \omega} _{\rm p}^2 /{\rm \omega} ^2.$ The amplitude of electric field E _± in laser beam is given by [Eq. (1)] which satisfies [Eq. (4)]. Under slowly varying envelope approximation, the evolution of electric field envelope in collisionless magnetized plasma can be written as,

(5)$$ \! \eqalign{- 2ik_ \pm \displaystyle{{\partial E_{0 \pm}} \over {\partial z}} + {\rm \delta} _ \pm \left( {\displaystyle{{\partial ^2E_{0 \pm}} \over {\partial r^2}} + \displaystyle{1 \over r}\displaystyle{{\partial E_{0 \pm}} \over {\partial r}}} \right) \! + \displaystyle{{{\rm \omega} ^2} \over {c^2}}{\rm \Phi} _ \pm (E_ \pm E_ \pm ^{^\ast} )E_{0 \pm} = 0.}$$

In WKB approximation, one can neglect ∂²E _±/∂z ² from [Eq. (4)]. The complex amplitude of electric vector may be expressed as,

(6)$$E_{0 \pm} (r,z) = A_ \pm (r,z)\exp [ - ik_ \pm S_ \pm (r,z)],$$

where A _±(r, z) and S _±(r, z) are the real functions of r and z, S _± is eikonal. Substituting [Eq. (6)] in [Eq. (5)] and separating into real and imaginary parts,

(7)$$\eqalign{& 2\displaystyle{{\partial S_ \pm} \over {\partial z}} + {\rm \delta} _ \pm \left( {\displaystyle{{\partial S_ \pm} \over {\partial r}}} \right)^2 + \displaystyle{{2S_ \pm} \over {k_ \pm}} \displaystyle{{\partial k_ \pm} \over {\partial z}} = \displaystyle{{{\rm \delta} _ \pm} \over {k_ \pm ^2 A_ \pm}} \cr & \;\left( {\displaystyle{{\partial ^2E_{0 \pm}} \over {\partial r^2}} + \displaystyle{1 \over r}\displaystyle{{\partial E_{0 \pm}} \over {\partial r}}} \right)A_ \pm + \displaystyle{{{\rm \omega} ^2} \over {c^2}}{\rm \Phi} _ \pm (A_ \pm A_ \pm ^{^\ast} )} $$

and

(8)$$\eqalign{& \displaystyle{{\partial A_ \pm ^2} \over {\partial z}} + {\rm \delta} _ \pm \displaystyle{{\partial S} \over {\partial r}}\displaystyle{{\partial A_ \pm ^2} \over {\partial r}} + {\rm \delta} _ \pm A_ \pm ^2 \left( {\displaystyle{{\partial ^2S_{0 \pm}} \over {\partial r^2}} + \displaystyle{1 \over r}\displaystyle{{\partial S_{0 \pm}} \over {\partial r}}} \right) \cr & \; + \displaystyle{{A_ \pm ^2} \over {k_ \pm}} \displaystyle{{\partial k_ \pm} \over {\partial z}} = 0.} $$

The solution of [Eqs. (7) and (8)] for cosh-Gaussian laser beam can be written as

(9a)$$\eqalign{A_{0 \pm }^2 &= \displaystyle{{E_{0 \pm }^2 } \over {{\mkern 1mu} f_ \pm ^2 }}\exp \left( {\displaystyle{{b^2} \over 2}} \right)\left\{ {\exp \left[ { - 2{\left( {\displaystyle{r \over {{\mkern 1mu} f_ \pm r_0}} + \displaystyle{b \over 2}} \right)}^2} \right]} \right. \cr & \left. { + \exp \left[ { - 2{\left( {\displaystyle{r \over {{\mkern 1mu} f_ \pm r_0}} - \displaystyle{b \over 2}} \right)}^{2}} \right] + 2\exp \left[ {{\left( {\displaystyle{{2r} \over {{\mkern 1mu} f_ \pm r_0}} + \displaystyle{b \over 2}} \right)}^2} \right]} \right\}} $$

(9b)$$S_ \pm = \displaystyle{{r^2} \over 2}{\rm \beta} _ \pm (z) + {\rm \varphi} _ \pm (z),$$

(9c)$${\rm \beta} _ \pm (z) = \displaystyle{1 \over {{\rm \delta} _ \pm}} \displaystyle{1 \over {\,f_ \pm}} \displaystyle{{df_ \pm} \over {dz}},$$

where $E_{0 \pm} ^2 $ is an initial laser intensity and f _± is the dimensionless beam-width parameter for extraordinary and ordinary modes, respectively. The ${\rm \beta} _ \pm ^{ - 1} $ is the radius of the curvature of wave front, φ_±(z) is the phase shift, and f _± is the beam-width parameter, which is the measure of both axial intensity and width of the beam. Substituting [Eqs. (9a)–(9c)] in [Eq. (7)] and equating coefficients of r ² on both sides we get

(10)$$\eqalign{& \displaystyle{{\partial ^2f_ \pm } \over {\partial \xi _ \pm ^2 }} = \displaystyle{1 \over {\varepsilon _{0 \pm }}}\displaystyle{1 \over {{\mkern 1mu} f_ \pm ^3 }}\left( {\displaystyle{{(12 - 12b^2 - b^4)\delta _ \pm ^2 } \over 3}} \right. \cr & \left. { - \displaystyle{{(2 - b^2)\exp [ - p_ \pm ]p_ \pm \rho _ \pm ^2 \gamma _ \pm \delta _ \pm } \over 2}} \right),} $$

where

$${\rm \gamma} _ \pm = \displaystyle{{{\rm \Omega} _{\rm p}^2} \over {1 \mp {\rm \Omega} _{\rm c}}},$$

$${\rm \Omega} _{\rm p} = \displaystyle{{{\rm \omega} _{\rm p}} \over {\rm \omega}}, $$

$${\rm \Omega} _{\rm c} = \displaystyle{{{\rm \omega} _{\rm c}} \over {\rm \omega}}, $$

$$p_ \pm = \displaystyle{{{\rm \alpha} E_{0 \pm} ^2} \over {\,f_ \pm ^2}}, $$

$${\rm \rho} _ \pm = \displaystyle{{r_0{\rm \omega}} \over c}.$$

The quantity p _± which is proportional to $E_{0 \pm} ^2 $ represents the beam power. The beam-width parameters $f_ \pm ^{} $ are the functions of ${\rm \xi} _ \pm ^{} $, with ${\rm \xi} _ \pm = z/k_ \pm r_0^2 $ as the normalized propagation distance.

Results and discussion

Equation (10) is a second-order, non-linear differential equation which represents variation of beam-width parameter f _± with normalized distance of propagation ξ _±. The first term on the right-hand side of [Eq. (10)] corresponds to the diffraction divergence of the beam and the second term corresponds to the convergence resulting from the ponderomotive non-linearity.

By subjecting [Eq. (10)] under critical condition (f _± = 1, ξ _± = 0), the general power of laser beam p _± and beam radius ${\rm \rho} _ \pm ^{} $ are replaced by critical power p _0± and critical beam radius ${\rm \rho} _{0 \pm} ^{} $. Therefore, right-hand side of [Eq. (10)] takes the form

(11)$$\eqalign{ F({\mkern 1mu} p_{0 \pm }) = & \displaystyle{{(12 - 12b^2 - b^4)\delta _ \pm ^2 } \over 3} \cr & - \displaystyle{{(2 - b^2)\exp [ - p_{0 \pm }]p_{0 \pm }\rho _{0 \pm }^2 \gamma _ \pm \delta _ \pm } \over 2}.} $$

Henceforth following investigation is restricted only for extraordinary mode.

(12)$$\eqalign{F({\mkern 1mu} p_{0 + }) = & \displaystyle{{(12 - 12b^2 - b^4)\delta _ + ^2 } \over 3} \cr & - \displaystyle{{(2 - b^2)\exp [ - p_{0 + }]p_{0 + }\rho _{0 + }^2 \gamma _ + \delta _ + } \over 2}.} $$

The condition that the critical beam power should be always finite and positive leads to the numerical interval of decentered parameter as 0 ≤ b ≤ 0.9634. By reducing defining equations for δ₊, γ ₊ and ${\rm \rho} _{0 +} ^{} $ numerically with the help of values N ₀ = 1 × 10¹⁸cm⁻³, ω = 1.776 × 10¹⁵rad/s, B ₀ = 10⁶gauss, r = 20 × 10⁻⁴cm and by defining the values of decentered parameter b = 0.00, 0.45, 0.90. [Eq. (12)] depends upon critical beam power.

To explore the effect of critical beam power right at the beginning one has to pay little attention to the plot shown in Figure 1. From the plot, it is clear that at the beginning, two values of critical beam powers p _0+lower and p _0+upper are possible.

Fig. 1. Variation of F(p ₀₊) with p ₀₊.

The function F(p ₀₊) has minimum value at p ₀₊ = 0.98, which is a common value for all decentered parameters ranging from 0.00 to 0.90. Hence, the value of p ₀₊ <0.98 and p ₀₊ >0.98 can be considered as p _0+lower and p _0+upper, respectively.

The plot shown in Figure 1 can be studied for three distinct conditions stated below and the simple analytical approach leads to the following limits for critical beam power.

For self-trapping:

$$\eqalign{F(\,p_{0 +} ) = & 0\;{\rm for}:p_{0 + {\rm lower}} = {\rm 0}{\rm. 00333927}\;{\rm and}\;{\rm} p_{0 + {\rm upper}} = {\rm 7}{\rm. 75349} (b = 0.00) \cr & \quad {\rm and}\;{\rm for}:p_{0 + {\rm lower}} = {\rm 0}{\rm. 00294922}\;{\rm and}\;{\rm} p_{0 + {\rm upper}} = {\rm 7}{\rm. 89545} (b = 0.45) \cr & \quad {\rm and}\;{\rm for}:p_{0 + {\rm lower}} = {\rm 0}{\rm. 000757512}\;{\rm and}\;p_{0 + {\rm upper}} = {\rm 9}{\rm. 43014} (b = 0.90).} $$

For defocusing:

$$\hskip-3pt F(\,p_{0 +} ) \gt 0\;{\rm for}:p_{0 + {\rm lower}} \lt {\rm 0}{\rm. 000757512}\;{\rm and} \; p_{0 + {\rm upper}} \gt 9.43014.$$

For self-focusing:

$$F(\,p_{0 +} ) \lt 0\,{\rm for}:(\,p_{0 + {\rm lower}}){\rm 0}{\rm. 00333927} \lt p_{0 +} \lt 7.75349\,(\,p_{0 + {\rm upper}}).$$

In the present analysis, authors are interested only in the self-focusing region in Figure 1.

From Figure 1, it is also seen that for above limits of critical beam powers for the condition of self-focusing of laser beams, three curves for three distinct values of decentered parameters intersect each other at three distinct points close to p _0+lower and p _0+upper as shown in insets of Figure 1.

In the analytical investigation, the intersecting points are called as turning points, which affect the propagation of laser beams through collisionless magnetized plasma. The turning points are as follow:

$$\eqalign{& p_{0 + {\rm lower}} = {\rm 0}{\rm. 00681507}\;(A)\;{\rm and}\;p_{0 + {\rm upper}} = {\rm 6}{\rm. 93151}\;(X)\;{\rm for}\;b = 0.00\;{\rm and}\;b = 0.45 \cr & p_{0 + {\rm lower}} = {\rm 0}{\rm. 00715664}\;(B)\;{\rm and}\;p_{0 + {\rm upper}} = {\rm 6}{\rm. 87472}\;(Y)\;{\rm for}\;b = 0.00\;{\rm and}\;b = 0.90 \cr & p_{0 + {\rm lower}} = {\rm 0}{\rm. 00727055}\;(C)\;{\rm and}\;p_{0 + {\rm upper}} = {\rm 6}{\rm. 85637}\;(Z)\;{\rm for}\;b = 0.45\;{\rm and}\;b = 0.90.} $$

The above study shows that preconditioning of a critical beam power in the beginning of propagation of laser beam can determine propagation dynamics effectively.

In the lower region, (p ₀₊ <0.98) of Figure 2a, 2b, which is the present region of interest, gives variation of beam-width parameter f ₊ against normalized propagation distance ξ ₊. In Figure 2a, 2b, strong self-focusing is observed. However, in Figure 2a, periodicity in self-focusing length is observed over larger interval of ξ ₊ as compared with periodicity in self-focusing length observed in Figure 2b over a given range of ξ ₊. Additionally, with increase in the values of decentered parameter, the effect of decentered parameter b on self-focusing length in Figure 2a is exactly opposite to that in Figure 2b. From the insets of Figure 2a, 2b, it is evident that the rate of self-focusing is exactly opposite with increase in the values of decentered parameter b.

Fig. 2. (a) Variation of beam-width parameter f ₊ with dimensionless propagation distance ξ ₊ before turning point for p _0+lower = 0.0035. (b) Variation of beam-width parameter f ₊ with dimensionless propagation distance ξ ₊ after turning point for p _0+lower = 0.15.

In the upper region, (p ₀₊ >0.98) of Figure 3a, 3b, which is the subsequent region of interest, gives variation of beam-width parameter f ₊ against normalized propagation distance ξ ₊. In Figure 3a, 3b, weak self-focusing is observed and the effect of decentered parameter b on self-focusing length is same, that is, with increase in the values of decentered parameter self-focusing, the length increases. Moreover, from Figure 3a, it is observed that with increase in the values of decentered parameter b, the rate of self-focusing is exactly opposite to that observed in Figure 3b, which is also evident from the insets of Figure 3a, 3b.

Fig. 3. (a) Variation of beam-width parameter f ₊ with dimensionless propagation distance ξ ₊ before turning points for p _0+upper = 5.5. (b) Variation of beam-width parameter f ₊ with dimensionless propagation distance ξ ₊ after turning point for p _0+upper = 7.2.